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"Physically impossible"? Mathworld has an illustration of a digon, along with a very different definition: "The digon is the degenerate polygon (corresponding to a line segment) with Schläfli symbol { 2 } ." Also, Dr. Micah Fogel at the Illinois Mathematics and Science Academy defines digons (and monogons) as "two special curved polygons that have no analogs among polygons with straight edges." Clearly a straight-edged digon would be physically impossible -- but do polygons necessarily have to have straight edges? Could two points connected by two curved lines qualify as a digon? The University of Wisconsin Oshkosh has a website discussing, "the two-sided polygon called a lune," including illustrations on how to calculate the area of such two-sided polygons. I'm not a mathematician, but after just five minutes of google searches I've found at least three sources that seem to be at-odds with the wikipedia definition. Could someone more knowledgeable about geometry expand (and correct, if need be) this article? 16:54, 6 June 2006 (UTC)

Done. Ages ago. Double sharp (talk) 14:53, 8 August 2009 (UTC)

regular digon?Edit

I don't think it makes sense to distinguish a regular digon. That is to say all digons on the sphere are regular since the edges must be great circles and two nonparallel great circles must intersect on two opposite points.

If there's no opinions, I'll remove the regular claim. Tom Ruen 01:23, 21 December 2006 (UTC)

Small correction perhaps. A digon can exist in a degenerate form on a sphere, just like in a degenerate form on the plane. So the only NONDEGENERATE digon on the sphere exists with polar vertices. Tom Ruen 01:27, 21 December 2006 (UTC)

Okay, I got creative, expanded an explantion of digons in polyhedra for Wythoff constructions. It might get a little off topic, but it was a pretty short article with just a definition! Tom Ruen 02:12, 21 December 2006 (UTC)

Oh no!Edit

Didn't we all agree to let digons be bygones? McKay 05:40, 11 January 2007 (UTC)

Digon imagesEdit

Someone left this in the main article. I don't really know what to do with it other than shunt it here, perhaps...

"I'm confused. What does a digon look like exactly?" (talk) 08:29, 13 March 2010 (UTC)

The digon is not degenerate in Euclidean GeometryEdit

If a circle is not degenerate then neither is a digon. If a digon was degenerate then why can you project a 3d digon on a 2d screen? Other shapes with two sides can be found in Euclidean geometry, a crescent for example two sides. —Preceding unsigned comment added by (talk) 14:27, 10 July 2010 (UTC) I think it means all straight lines (but that is highly redundant, that should mean circles are degenerate shapes they are not) We should link to Vesica piscis.-- (talk) 15:31, 27 March 2013 (UTC)

No, the point is that a digon with straight lines is degenerate. With curves it is not really a polygon (and hence not really a digon). Double sharp (talk) 15:53, 27 March 2013 (UTC)
It's a matter of geodesics to define "straight lines". In Euclidean space, there's only a single minimum length path (a straight line) between two points, so a digon can be defined only if it is degenerate. However on curved surfaces, like a sphere or torus, there can be two or more different minimum length paths, and so the digon can be defined with two different edge paths. But even on a sphere, two arbitrary points won't be antipodal, and so will also be degenerate. Tom Ruen (talk) 17:23, 27 March 2013 (UTC)

Digons in Wythoff constructionEdit

Here's an example graphic (right) I made to demonstrate the nature of digons in a Wythoff construction on a circle. The symbol {2} representsCoxeter diagram for the digon, as     or    .

So some where we have to explain each node represents a mirror, and these two mirrors are orthogonal, usually with an implicit branch order 2 by no connection. The ring represents an active mirror where there's a virtual image of the generating point across the mirror, while a ringless node is inactive, so the generating point must be on the mirror.

The related Coxeter group, D2, or [2] has an unringed Coxeter diagram     or     or     where   is the vertical mirror (green), and   is the horizontal mirror (cyan). The nodes can also be colored to match the lines of reflection in the diagram,    . A Coxeter diagram without circles or holes is assumed to be a Coxeter group because its imposible for all mirrors to be inactive (except if the generator point was at the center of the circle, which for tilings is not allowed.)

So     represents an active vertical mirror, with 2 vertices on the x-axis. And     represents an active horizontal mirror, so the 2 vertices are on the y-axis.     represent both active mirrors, which can make a "rectangle" in general but by convention is assumed to be the equilateral solution, thus becomes a "circular square". So I drew the square with 4 colors of edges, and you can see the digons represent degenerate "rectangles" where either set of edges reduces to zero length.

This sort of degeneracy exists in all of the Wythoff constructions, so a cube,       is a degenerate truncated cuboctahedron,      , with one active mirror, and two inactive mirrors, reducing faces into edges or vertices, and edges into vertices.


Finally we can add two final cases on the right most edge of the graphic: h{2},     as a "half digon", removing alternate vertices from {2} which becomes a monogon, and s{2},    , as a "snub digon", removing alternate vertices from t{2} or {4}, which becomes another digon, {2} in a different orientation.

So ALL of this is expressed in Coxeter's work, but he never bothers to explain these elements in the most basic cases like this.

So is it original research to take generalization and give explicit examples? If it is, if every fact expressed on Wikipedia has to be defended by explicit references, then we're not allowed to explain anything that sources don't explain well. Then there's no use in knowing anything, and we should just type verbatim what is written in books, or summarizing it, leaving examples out, unless such examples are given explicitly by a source.

And without detailed OR examples derived as simplest examples from a general theory, we can't ask "Why can't you have a digon constructed with two edge lengths, one long and short path?" We have to say "We don't know, because Coxeter didn't tell us and he's dead."

That's a rather poverty strickened system of teaching new understanding to readers. Tom Ruen (talk) 08:29, 8 January 2015 (UTC)

As a general principle, I would suggest that we can take a general comment about a degenerate situation and give illustrative examples. What we can not do is to take a general comment about a non-degenerate situation and give a degenerate example. For example the progressive transformations involving bevelling and truncation have been well described, so your illustration here should have no trouble in finding sources. That is to say, one can source a comment on the degeneracy of squares into digons and then present illustrative examples. But if one were to source a comment on digons as valid constructions and then present them as degenerate in Euclidean space, that would not be acceptable: One needs a reference to say that they are degenerate in Euclidean space. We can interpolate, but we cannot extrapolate.
One way of looking at the "teaching" issue is to recall WP:NOTTEXTBOOK. Wikipedia is an encyclopedia of verifiably existing knowledge, not of new knowledge. That need for WP:VERIFIABILITY can be hard for some folks to swallow, but it cannot be helped. If one's own thinking struggles to verify some aspect of what one is writing, then it is most likely that one has not fully appreciated what the sources are - and are not - saying.
For example the question, "Why can't you have a digon constructed with two edge lengths, one long and short path?" is actually wrong-headed. You can, for example as a step in the topological analysis of a smooth manifold. But it's a digon constructed to slightly different rules. Without being clear about the constraints placed on the topic by the author, one can be caught out. Of course, geometers can be notoriously sloppy about getting their boundary conditions clear before they start, so that gives us encyclopedists a double-challenge in presenting verifiable knowledge that happens to be inconsistent - but seldom verifiably so. One must be cautious. One can interpolate where the bounding assumptions are clear, but one cannot extrapolate beyond any clear boundary. This is especially so when close to a supposedly strict boundary where the authorities' unconscious assumptions, for example that a digon always has equal sides, might be invalid.
Once we know what can be said, then we are faced with how to say it. For example, should we say it once and scatter around links to that article? Or, should we repeat it over and over on every darn article and in every darn image that it is relevant to? Or, maybe throw everything it is relevant to into one giant article? Excessive brevity and link-clicking get unreadable and tiresome, excessive clutter and page length get unreadable and tiresome, there is a balance to be struck, but it is a different argument from what can be said.
Poverty may arise where the sources fail to give a consistent picture, or where we ourselves muddle up zeal for verifiability with zeal for presentation. But poverty is better than misinformation. — Cheers, Steelpillow (Talk) 11:51, 8 January 2015 (UTC)

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